Wrapped Fukaya categories and functors

Inspired by homological mirror symmetry for non-compact manifolds, one wonders what functorial properties wrapped Fukaya categories have as mirror to those for the derived categories of the mirror varieties, and also whether homological mirror symmetry is functorial. Comparing to the theory of Lagrangian correspondences for compact manifolds, some subtleties are seen in view of the fact that modules over non-proper categories are complicated. In this talk, the story concerning the fundamental construction of Fourier-Mukai type functors of wrapped Fukaya categories is discussed, under slightly modified framework of wrapped Floer theory. Applications of the relevant techniques to be presented include the Kunneth formula and restriction maps.